Check if a Tree can be split into K equal connected components

Given Adjacency List representation of a tree and an integer K., the task is to find whether the given tree can be split into K equal Connected Components or not.
Note: Two connected components are said to be equal if they contain equal number of nodes.

Examples: 

Input: N = 15, K = 5 
Beow is the given tree with Number nodes = 15 
 

Output: YES 
Explanation: 
Below is the 5 number of Connected Components can be made: 
 



Approach: 
The idea is to use Depth First Search(DFS) traversal on the given tree of N nodes to find whether the given tree can be split into K equal Connected Components or not. Following are the steps: 

  1. Start DFS Traversal with the root of the tree.
  2. For every vertex not visited during DFS traversal, recursively call DFS for that vertex keeping the count of nodes traverse during every DFS recursive call.
  3. If the count of nodes is equals to (N/K) then we got our one of the set of Connected Components.
  4. If the total number of the set of Connected Components of (N/K) nodes is equal to K. Then the given graph can be split into K equals Connected Components.

Below is the implementation of the above approach: 

C++

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// C++ program to detect whether
// the given Tree can be split
// into K equals components
#include <bits/stdc++.h>
using namespace std;
 
// For checking if the graph
// can be split into K equal
// Connected Components
int flag = 0;
 
// DFS Traversal
int DFS(vector<int> adj[], int k,
        int i, int x)
{
 
    // Intialise ans to 1
    int ans = 1;
 
    // Traverse the adjacency
    // for vertex i
    for (auto& it : adj[i]) {
        if (it != k) {
            ans += DFS(adj, i, it, x);
        }
    }
 
    // If number of nodes is
    // greater than x, then
    // the tree cannot be split
    if (ans > x) {
        flag = 1;
        return 0;
    }
 
    // Check for requirement
    // of nodes
    else if (ans == x) {
        ans = 0;
    }
    return ans;
}
 
// A utility function to add
// an edge in an undirected
// Tree
void addEdge(vector<int> adj[],
             int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}
 
// Driver's Code
int main()
{
    int N = 15, K = 5;
 
    // Adjacency List
    vector<int> adj[N + 1];
 
    // Adding edges to List
    addEdge(adj, 1, 2);
    addEdge(adj, 2, 3);
    addEdge(adj, 2, 4);
    addEdge(adj, 4, 5);
    addEdge(adj, 5, 6);
    addEdge(adj, 5, 7);
    addEdge(adj, 4, 8);
    addEdge(adj, 4, 9);
    addEdge(adj, 8, 11);
    addEdge(adj, 10, 11);
    addEdge(adj, 11, 14);
    addEdge(adj, 9, 12);
    addEdge(adj, 12, 15);
    addEdge(adj, 12, 13);
 
    // Check if tree can be split
    // into K Connected Components
    // of equal number of nodes
    if (N % K == 0) {
        // DFS call to Check
        // if tree can be split
        DFS(adj, -1, 1, N / K);
    }
 
    // If flag is 0, then the
    // given can be split to
    // Connected Components
    cout << (flag ? "NO" : "YES");
 
    return 0;
}

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Java

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// Java program to detect whether
// the given Tree can be split
// into K equals components
import java.util.*;
 
class GFG
{
  
// For checking if the graph
// can be split into K equal
// Connected Components
static int flag = 0;
  
// DFS Traversal
static int DFS(Vector<Integer> adj[], int k,
        int i, int x)
{
  
    // Intialise ans to 1
    int ans = 1;
  
    // Traverse the adjacency
    // for vertex i
    for (int it : adj[i]) {
        if (it != k) {
            ans += DFS(adj, i, it, x);
        }
    }
  
    // If number of nodes is
    // greater than x, then
    // the tree cannot be split
    if (ans > x) {
        flag = 1;
        return 0;
    }
  
    // Check for requirement
    // of nodes
    else if (ans == x) {
        ans = 0;
    }
    return ans;
}
  
// A utility function to add
// an edge in an undirected
// Tree
static void addEdge(Vector<Integer> adj[],
             int u, int v)
{
    adj[u].add(v);
    adj[v].add(u);
}
  
// Driver's Code
public static void main(String[] args)
{
    int N = 15, K = 5;
  
    // Adjacency List
    Vector<Integer> []adj = new Vector[N + 1];
    for(int i= 0; i < N + 1; i++)
        adj[i] = new Vector<Integer>();
     
    // Adding edges to List
    addEdge(adj, 1, 2);
    addEdge(adj, 2, 3);
    addEdge(adj, 2, 4);
    addEdge(adj, 4, 5);
    addEdge(adj, 5, 6);
    addEdge(adj, 5, 7);
    addEdge(adj, 4, 8);
    addEdge(adj, 4, 9);
    addEdge(adj, 8, 11);
    addEdge(adj, 10, 11);
    addEdge(adj, 11, 14);
    addEdge(adj, 9, 12);
    addEdge(adj, 12, 15);
    addEdge(adj, 12, 13);
  
    // Check if tree can be split
    // into K Connected Components
    // of equal number of nodes
    if (N % K == 0) {
        // DFS call to Check
        // if tree can be split
        DFS(adj, -1, 1, N / K);
    }
  
    // If flag is 0, then the
    // given can be split to
    // Connected Components
    System.out.print(flag==1 ? "NO" : "YES");
}
}
 
// This code is contributed by Rajput-Ji

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Python3

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# Python3 program to detect whether
# the given Tree can be split
# into K equals components
 
# For checking if the graph
# can be split into K equal
# Connected Components
flag = 0
 
# DFS Traversal
def DFS(adj, k, i, x):
     
    # Intialise ans to 1
    ans = 1
    
    # Traverse the adjacency
    # for vertex i
    for it in adj[i]:
        if it is not k:
            ans += DFS(adj, i, it, x)
             
    # If number of nodes is
    # greater than x, then
    # the tree cannot be split
    if (ans > x):
        flag = 1
        return 0
    
    # Check for requirement
    # of nodes
    elif (ans == x):
        ans = 0
     
    return ans
 
# A utility function to add
# an edge in an undirected
# Tree
def addEdge(adj, u, v):
     
    adj[u].append(v)
    adj[v].append(u)
 
# Driver code
if __name__=="__main__":
     
    (N, K) = (15, 5)
     
    # Adjacency List
    adj = [[] for i in range(N + 1)]
     
    # Adding edges to List
    addEdge(adj, 1, 2);
    addEdge(adj, 2, 3);
    addEdge(adj, 2, 4);
    addEdge(adj, 4, 5);
    addEdge(adj, 5, 6);
    addEdge(adj, 5, 7);
    addEdge(adj, 4, 8);
    addEdge(adj, 4, 9);
    addEdge(adj, 8, 11);
    addEdge(adj, 10, 11);
    addEdge(adj, 11, 14);
    addEdge(adj, 9, 12);
    addEdge(adj, 12, 15);
    addEdge(adj, 12, 13);
     
    # Check if tree can be split
    # into K Connected Components
    # of equal number of nodes
    if (N % K == 0):
         
        # DFS call to Check
        # if tree can be split
        DFS(adj, -1, 1, N // K)
    
    # If flag is 0, then the
    # given can be split to
    # Connected Components
    if flag == 1:
        print("NO")
    else:
        print("YES")
 
# This code is contributed by rutvik_56

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C#

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// C# program to detect whether
// the given Tree can be split
// into K equals components
using System;
using System.Collections.Generic;
 
class GFG
{
   
// For checking if the graph
// can be split into K equal
// Connected Components
static int flag = 0;
   
// DFS Traversal
static int DFS(List<int> []adj, int k,
        int i, int x)
{
   
    // Intialise ans to 1
    int ans = 1;
   
    // Traverse the adjacency
    // for vertex i
    foreach (int it in adj[i]) {
        if (it != k) {
            ans += DFS(adj, i, it, x);
        }
    }
   
    // If number of nodes is
    // greater than x, then
    // the tree cannot be split
    if (ans > x) {
        flag = 1;
        return 0;
    }
   
    // Check for requirement
    // of nodes
    else if (ans == x) {
        ans = 0;
    }
    return ans;
}
   
// A utility function to add
// an edge in an undirected
// Tree
static void addEdge(List<int> []adj,
             int u, int v)
{
    adj[u].Add(v);
    adj[v].Add(u);
}
   
// Driver's Code
public static void Main(String[] args)
{
    int N = 15, K = 5;
   
    // Adjacency List
    List<int> []adj = new List<int>[N + 1];
    for(int i= 0; i < N + 1; i++)
        adj[i] = new List<int>();
      
    // Adding edges to List
    addEdge(adj, 1, 2);
    addEdge(adj, 2, 3);
    addEdge(adj, 2, 4);
    addEdge(adj, 4, 5);
    addEdge(adj, 5, 6);
    addEdge(adj, 5, 7);
    addEdge(adj, 4, 8);
    addEdge(adj, 4, 9);
    addEdge(adj, 8, 11);
    addEdge(adj, 10, 11);
    addEdge(adj, 11, 14);
    addEdge(adj, 9, 12);
    addEdge(adj, 12, 15);
    addEdge(adj, 12, 13);
   
    // Check if tree can be split
    // into K Connected Components
    // of equal number of nodes
    if (N % K == 0) {
        // DFS call to Check
        // if tree can be split
        DFS(adj, -1, 1, N / K);
    }
   
    // If flag is 0, then the
    // given can be split to
    // Connected Components
    Console.Write(flag==1 ? "NO" : "YES");
}
}
 
// This code contributed by Rajput-Ji

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Output: 

YES

 

Time Complexity: O(V + E), where V is the number of vertices and E is the number of edges
 

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